Approximation sequence processing

ABSTRACT

A method of producing an approximation sequence to a series of sample values, the method comprising the steps of (a) determining a first set having candidate partial sequences as members, each member comprising a plurality of elements; (b) selecting the first n elements of one of the members of the first set as a next output element for said approximation sequence; n a positive integer; (c) forming a second set having descendent candidate partial sequences as members from said first set; (d) applying a fitness filtering process to said second set to rank its members according to fitness for representing at least a corresponding portion of the series of input samples; (e) selecting at least some of the members of the second set to form a third set; and repeating steps (a) to (e) so as to produce said approximation sequence, wherein the third set of step (e) functions as the first set of the subsequent step (a).

RELATED PATENT APPLICATIONS

This invention claims priority under 35 USC 119 of AustralianApplication No. Australian 2002950530, inventor David McGrath, filed onAug. 1, 2002, titled “Approximation Sequence Processing,” assigned tothe assignee of the present invention, and incorporated herein byreference.

BACKGROUND

1. Field of the Invention

The present invention relates broadly to a method and system forproducing a sequence approximating a series of sample values. The samplevalue series may be never-ending. The present invention will bedescribed herein with reference to a method and system for producing a1-bit sequence from samples of an input waveform.

2. Background of the Invention

The present invention has been developed after extensive studies in thefield of Sigma-Delta modulation. Sigma-delta modulators are normallyutilized to convert analog signals to sequences of corresponding 1-bitvalues such that the filtering of such a 1-bit sequence by a low passfilter results in a reproduction of the original analog signal or aclose version thereof. For a background description of the operation of1-bit sigma-delta modulators, reference is made to standard textbooks inthe field such as “Delta-Sigma Data Converters, Theory Design andSimulation” by Norsworthy, et al., IEEE press, 1997.

More specifically, the accurate approximation of an analog input signalby a sequence of 1-bit values wherein the bits take on the usual +1 and−1 values was initially investigated. The design being such that uponfiltering of the sequence of 1-bit values, a close approximation of theoriginal analog signal is returned.

Turning now to FIG. 1-4, there is illustrated an example of the steps ina normal 1-bit conversion process. In FIG. 1, there is illustrated anexample input waveform 10, with a maximum amplitude that is no greaterthan +/−P. The normal conversion process involves sampling the waveformof FIG. 1 at predetermined points in time separated by a time intervalso as to produce a series of sample values, e.g., 12 as illustrated inFIG. 2. Next, the core process of converting the sample values tocorresponding +1 and −1 values is illustrated in FIG. 3 with a samplevalue illustrated, e.g., 14. Subsequently, as illustrated in FIG. 4, aone-bit output stream is output 16 corresponding to the set of samplevalues determined in FIG. 3. The determination of the sample values canproceed by many prior art methods. The preferred embodiment of thepresent invention is directed to the creation of an improveddetermination method which produces lower relative levels of noise thanthat provided by prior art methods.

The maximum input signal amplitude relative to the maximum output signalamplitude, (P) must typically be less than 1, and in practice, mosttechniques used to produce one-bit output streams will require that P isno greater than approximately 0.5. A further preferred embodiment of thepresent invention is directed to the generation of one-bit sequenceswherein the maximum signal level (P) may be significantly greater than0.5, with typical values as high as 0.75 being achievable, without asignificant reduction of Signal to Noise Ratio (SNR).

SUMMARY

In accordance with a first aspect of the present invention, there isprovided a method of producing a sequence (an “approximation sequence”)to approximate a series of sample values, the method comprising thesteps of (a) determining a first set having candidate partial sequencesas members, each member comprising a plurality of elements; (b)selecting the first n elements, n being any positive integer value, ofone of the members of the first set as a next output element for saidapproximation sequence; (c) forming a second set having descendentcandidate partial sequences as members from said first set; (d) applyinga fitness filtering process to said second set to rank its membersaccording to fitness for representing at least a corresponding portionof the series of input samples;(e) selecting at least some of themembers of the second set to form a third set; and repeating steps (a)to (e) so as to produce said approximation sequence, wherein the thirdset of step (e) functions as the first set of the subsequent step (a).

In one embodiment, the method further comprises, between steps (b) and(c), the step of (b*) as follows:

-   -   (b*) deleting one or more members of said first set. Step (b*)        may comprise removing all members of the first set which do not        begin with the selected first n elements of step (b).

Preferably, step (e) is conducted in a manner such as to maintain apredetermined number of members in the first set for each iteration.

In one embodiment, each element of the approximation sequence has mpossible values, where m in an integer that can take on any valuegreater than 1, and step (c) comprises, for each member of the firstset, forming corresponding m^(n) descendent partial sequences as membersof the second set by appending each possible value/combination to andremoving the first n elements of the member.

In another embodiment, each element of the approximation sequence has mpossible values me being an integer greater than 1, and step (c)comprises, for each member of the first set, forming less than m^(n)corresponding descendent partial sequences as members of the second setby appending a limited number of possible value/combinations to andremoving the first n elements of the member.

In one embodiment, step (d) comprises one or more of a group comprisinglow pass filtering, finite impulse response filtering, and recursivefiltering each candidate partial sequence of the first set.

Advantageously, step (d) comprises giving highest precedence to membersof the first set which begin with an element having a same sign as thenext sampling point.

In one embodiment, the method further comprises the step of sampling awaveform signal to obtain the series of sample values.

Preferably, step (d) comprises applying the fitness filtering process tothe members of the second set such as to rank them according to fitnessfor approximating the series of sample values from the first samplevalue up to and including the next sample value to be approximated.

In one embodiment, the output elements are output in the form ofsingle-bit or multi-bit samples.

In accordance with a second aspect of the present invention, there isprovided a system for producing a sequence (an “approximation sequence”)to approximate a series of sample values, the system comprising aprocessing unit arranged, in use, to (a) determine a first set havingcandidate partial sequences as members, each member comprising aplurality of elements; (b) select the first n elements—n being apositive integer—of one of the members of the first set as a next noutput elements for said approximation sequence; (c) form a second sethaving descendent candidate partial sequences as members from said firstset; (d) apply a fitness filtering process to said second set to rankits members according to fitness for representing at least acorresponding portion of the series of input samples; (e) select atleast some of the members of the second set to form a third set; acontrol unit arranged, in use, such that the processing unit repeatssteps (a) to (e), wherein the third set of step (e) functions as thefirst set of the subsequent step (a); and an output unit arranged, inuse, to output the selected at least one output element during eachiteration, whereby the approximation sequence is produced.

In one embodiment, the processing unit is further arranged, in use, to,between steps (b) and (c), (b*) delete one or more members of said firstset.

Preferably, step (b*) comprises removing all members of the first setwhich do not begin with the selected at least first element of step (b).

Advantageously, the processing unit is arranged, in use, to conduct step(e) in a manner such as to maintain a predetermined number of members inthe first set for each iteration.

In one embodiment, each element of the approximation sequence has mpossible values, m an integer greater than 1, and step (c) comprises,for each member of the first set, forming corresponding m^(n) descendentpartial sequences as members of the second set by appending eachpossible value/combination to and removing the first n elements of themember.

In another embodiment, each element of the approximation sequence has mpossible values, m an integer greater than 1, and step (c) comprises,for each member of the first set, forming less than m^(n) correspondingdescendent partial sequences as members of the second set by appending nelements with a limited number of possible values/combinations to andremoving the first n elements of the member.

Preferably, step (d) comprises one or more of a group comprising lowpass filtering, finite impulse response filtering, and recursivefiltering each candidate partial sequence of the first set.

In one embodiment, step (d) comprises giving highest precedence tomembers of the first set which begin with an element having a same signas the next sampling point.

Preferably, the system further comprises a sampling unit for sampling awaveform signal to obtain the series of sample values.

Advantageously, step (d) comprises applying the fitness filteringprocess to the members of the second set such as to rank them accordingto fitness for approximating the series of sample values from the firstsample value up to and including the next n sample values to beapproximated.

In one embodiment, the output unit is arranged, in use, such that theoutput elements are output in the form of single-bit or multi-bitsamples.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the invention will now be described, by way ofexample only, with reference to the accompanying drawings in which:

FIG. 1 illustrates a time domain signal to be converted to acorresponding 1-bit signal;

FIG. 2 illustrates the sample equivalent of the signal of FIG. 1;

FIG. 3 illustrates an example equivalent 1-bit sequence;

FIG. 4 illustrates an output 1-bit series corresponding to that of FIG.3;

FIG. 5 illustrates the process of a preferred embodiment;

FIG. 6 illustrates a detail of the process of a preferred embodiment;

FIG. 7 is a schematic illustration of the process of determiningcandidate children, embodying the present invention;

FIG. 8 illustrates a more specific example of the preferred embodiment;

FIG. 9 illustrates the processing applied to the example of FIG. 8, asper the preferred embodiment;

FIG. 10 illustrates a method for determining the fitness of a candidateembodying the present invention;

FIG. 11 illustrates a variation of the process shown in FIG. 9, as per afurther preferred embodiment;

FIGS. 12A-12C illustrate variants of each of FIGS. 1-3, respectively, inwhich the output samples are restricted to 4 allowable quantizationlevels; and

FIG. 13 illustrates an alternative embodiment where the number ofallowable quantization levels, m, is large.

DETAILED DESCRIPTION

In the preferred embodiment, as noted previously, an improved method isprovided for determining a sequence we call an approximation sequencethat approximates a corresponding series of sample values. In thepreferred embodiment, the approximation sequence is a 1-bit outputstream that approximates a corresponding multi-bit sampled input stream.Hence, the problem can be generally stated that given a set of samplevalues such as those illustrated in FIG. 2, what form of output streamsuch as that shown in FIGS. 3 and 4 should be produced.

As illustrated in FIG. 5 the preferred embodiment solves that problem byutilizing a partial sequence window 40 containing a certain number ofelements (eight, in this example), wherein the first element of thepartial sequence window 40 is utilized as the next output element e.g.42, and subsequently the partial sequence window 40 is moved by oneelement, as illustrated by arrow 44 in FIG. 5.

The preferred embodiment thus proceeds by outputting elements one at atime to an output stream. The preferred embodiment generates a series ofoutput stream candidates and ranks them to determine which candidate isto be utilized. In FIG. 6 there is illustrated a series of candidatepartial sequences in the form of bit strings 50-53. The structure of thecandidate output streams is characterized by two parameters. The firstis a history parameter 54 which defines the length of the candidatestreams as illustrated in FIG. 6. The history e.g., 54 defines thelength of the processing window of candidate 50 utilized in thepreferred embodiment. The number of candidate streams 50-53 (alsoreferred to as the population size) is a second parameter and, in theexample illustrated in FIG. 6, is equal to 4 with the history beingequal to 8 elements. At each iteration, the first element of the fitteststring is output e.g., 58 and the window 50 “moved along”. In theembodiment of FIG. 6, there are illustrated the four possible candidates50-53 at this stage of processing.

After the output of an element, e.g. 58 as part of the “build-up” of theoutput sequence, some candidates that do not match the output bit e.g.58 may now be deleted (e.g. 52, 53). The process proceeds by proposingtwo possible descendants for each of the remaining candidates (e.g. 50,51). For example, in FIG. 7 there is illustrated the process of takingthe first candidate 50 of FIG. 6 and proposing two descendent candidates60, 62 with the candidates 60, 62 being created by appending a +1 datavalue 68 and a −1 data value 70 respectively, and removing the firstelement 58 of candidate 50.

The number of descendent candidates is preferably maintained at theoriginal number of candidates for the subsequent approximationiteration, through appropriate culling before or after the creation ofthe descendants. For example, in the arrangement of FIG. 6, there were 4candidates 50-53 in the original group. The first output bit 58 of thehighest ranking candidate 50 was output. The descendent population isculled to only include those candidate sequences having a matching bitto the chosen most fittest candidate, in the example shown candidates 50and 51. Further culling routines and criteria, preferably in order offitness, can be provided for in the process to arrive at the desirednumber.

Thus, 4 descendants 60, 62, 64 and 66 shown in FIG. 7 become the “next”possible candidates. Those candidates not having a matching firstelement may be removed from the candidate list. The next possiblecandidates are subjected to a fitness filtering process to determinewhich candidates are most suitable. This proceeds by ranking them in afitness order. The fitness ranking is then used to select the bestcandidates for the next iteration of the process. Further culling caninclude applying a filter such as a finite impulse response filter or arecursive filter to the candidate strings and determining a closestmatch to the corresponding original input signal portion (e.g. as shownin FIG. 2). The most suitable candidate, e.g. 62, is then determinedhaving the closest characteristics to the input signal, and its firstelement 72 is output.

Turning of FIG. 8, there is illustrated an example wherein a sequence ofoutput elements has already been determined, and a population of 4candidates A, B, C and D, each with a history length of 4, have beenselected (by previous iterations of the process).

Turning to FIG. 9, there is illustrated a general flow chart of thesteps involved in one iteration of the method in the example embodiment,based on the set of candidates A, B, C and D shown in FIG. 8. Theexample embodiment is an iterative method, and starts with the selectionof a new output sample, by either (a) choosing the first sample of the‘fittest’ member of the candidate population 130 (A, B, C, or D) or (b)choosing the most ‘popular’0 first sample. In the example of FIG. 9, theselected output sample 131 is −1, chosen because it occurs in a greaterproportion of the population 130 i.e. it is most popular. If −1 and +1occur in the first sample of the candidates in equal quantities, thensome added criteria is required to select the new output sample. Thismay include the choice of the first sample from the candidate with thehighest ‘fitness’ score, or may include the use of an arbitraryselection. Once the new output element has been determined, thecandidate population is culled to an intermediate set 132 by removal ofall candidates that do not match the selected output element in theexample candidate D. A pair of descendants is generated for each memberof the culled population, by removing the first sample, and appending −1or +1. The set of new descendants 134 thus comprise A1, A2, B1, B2, C1and C2 in the shown example. Next, the fitness of each candidate iscomputed, and the four fittest candidates are maintained in a final set136, in the example shown A1, B1, B2 and C2, ready for the nextiteration of the method.

FIG. 10 illustrates the method by which the fitness of each candidate isdetermined in the example embodiment. Even though a candidate A1 has arelatively short length (the history length), the fitness function isbased on the augmented sequence, formed by appending the candidatesequence A1 onto the end of the current output sequence 100. The inputsequence 102—including a new input sample 104 that was not used in theprevious iteration of the method—is subtracted from the augmentedsequence, and the difference signal is filtered to produce a filterederror measure that has frequency dependent weighting. The magnitude ofthe frequency weighted error signal is used as the negative of thefitness measure to be maximized, e.g., the candidate or subset isselected that has the largest fitness measure, i.e., the smallestfiltered error measure magnitude.

In FIG. 11, an alternative example embodiment is illustrated, showing amodified fitness selection method. In the embodiment of FIG. 11, thecandidate population is forced to always have the same value in thefirst sample of each candidate, thus ensuring that no culling of thepopulation is required prior to the generation of the descendants. Thus,in FIG. 11 after extraction of the output sample 120 on the basis of aset of candidates 122, descendants are formed for each of the members ofthe set of candidates 122 to form the intermediate set of descendants124. No candidates were culled from the set 122 because the set wasguaranteed to always contain candidates with identical initial values.The method by which the next generation of candidates is selected fromthe set of descendants 124 is adapted to ensure that the set ofdescendants is chosen such that the fittest set of four are selectedwith the proviso that all four must have the same value in their firstsample. Thus, in FIG. 11 fitness functions are computed and thedescendants 124 ranked in order, to form a ranked set of descendants126. In the example embodiment, the way of ensuring that this criteriais satisfied, is to select the fittest four descendants that start with+1, and the fittest four descendants that start with −1. If either ofthese sets contains less than 4 members, for example, if there are lessthan 4 descendants starting with +1, as in the example shown in FIG. 11,then that set will be culled. Of the un-culled sets, the fittest isselected based any of a number of different criteria, including (a)selecting the set with the highest average fitness, (b) selecting theset which contains the fittest overall member, or (c) selecting the setwith the worst (fourth) member having the highest fitness relative tothe worst (fourth) member of the other sets. In the example shown inFIG. 11, this leaves D1, B2, C1 and C2 as the next set of candidates128.

In alternative preferred embodiments, the method of the presentinvention may be utilized to generate quantized approximation sequenceswhere the number of allowable output values m for each element is morethan 2. For example, referring to FIG. 12(c), if each element has 4allowable values, rather than the 1-bit values used in the embodimentsdescribed above, then each output sample can take on any one of 4possible values. Looking again at FIG. 11, it will be clear to thoseskilled in the art that the method illustrated in FIG. 11 can beextended to allow for this larger set of allowable output values. Inthis case, the expanded population 124 of new candidate sequences willbe 4 times larger than the original population 122.

It is further noted that the values for each element itself may beoutput as a multi-bit output sample, e.g. 16-bit audio samples forrecording onto a Compact Disc, rather than each element being output asa 1-bit sample as in the embodiments described above. In such anembodiment, each element has effectively 2 ¹⁶ allowable “values”. It maybe impractical to test all 65536 possible descendents in such anembodiment, as this will lead to a population of e.g. 4 initialcandidates (122) growing to a new set of 262144 new dependants (124)(referring by analogy to FIG. 11). Thus, in a further preferredembodiment, a small sub-set of all potential output samples is selected,so that each candidate will then spawn only a relatively small number ofdependants, thus maintaining the population at a more manageable level.

This method is illustrated in FIG. 13, in which an original input signal151 is approximated by a series of quantized sample values, e.g., 156.Each sample value is constrained to lie on one of m quantization levels,e.g., 155. In the example of FIG. 13, the output sample sequence 160 hasbeen determined, and one of the next candidates is sequence (A). Themethod of generating the new descendants (A1, A2, A3) from thiscandidate (A) is carried out, in an example embodiment, by firstselecting three allowable values for the next element out of thepossible m values. Whilst the number of allowable quantization values mmay be very large, in the example we start with a reasonable estimatequantization level 153 for the next sample, and add it's immediateneighbors 152 and 154, to form a set of three alternatives. Hence, ourcandidate population will grow by a factor of three each time thedescendant candidates are generated.

Furthermore, the method of the present invention may be utilized togenerate quantized approximation sequences where the number n of outputelements for each output “iteration” is more than 1. For example, if 2elements are output at each iteration, rather than the one element as inthe embodiments described above, then the ‘sliding window’ (compare 40in FIG. 5) is moved forward by 2 elements in each iteration of themethod.

It was found that, when utilizing the aforementioned example methods forthe creation of 1-bit output streams, the corresponding noise floor ofthe 1-bit output stream was substantially lowered leading to improvedresults. Hence, the aforementioned example methods are able to produceimproved 1-bit sequences. In addition, is was found that the maximuminput signal level P (FIG. 1) could be made larger when utilizing theaforementioned example methods, in contrast to alternative prior artimplementations that can fail to operate correctly when presented withinput signals of excessive magnitude.

It would be evident to those skilled in the art that the preferredembodiment uses a search method that may have advantages over using aprior art searching procedure, e.g., a prior art MinMax procedure inthat a MinMax procedure often suffers from exponential complexity growthproblems. The utilization of candidate pruning allows for pruning tobranches of a tree whilst simultaneously maintaining many possiblealternative end solutions. The utilization of the history allows forsome form of depth first searching to take place whilst simultaneouslylimiting the number of branches at each node.

The aforementioned methods can be programmed into a computer system suchas a personal computer type system so as to process input signals toproduce 1-bit output signals for use.

It will be appreciated by the person skilled in the art that numerousmodifications and/or variations may be made to the present invention asshown in the specific embodiments without departing from the spirit orscope of the invention as broadly described. The present embodimentsare, therefore, to be considered in all respects to be illustrative andnot restrictive.

In the claims that follow and in the summary of the invention, exceptwhere the context requires otherwise due to express language ornecessary implication the word “comprising” is used in the sense of“including”, i.e. the features specified may be associated with furtherfeatures in various embodiments of the invention.

1. A method of producing a sequence approximating a series of samplevalues, the method comprising the steps of: (a) determining a first sethaving candidate partial sequences as members, each member comprising aplurality of elements; (b) selecting the first n elements of one of themembers of the first set as the next n output elements for saidapproximation sequence, n being a positive integer; (c) forming a secondset having descendent candidate partial sequences as members from saidfirst set; (d) applying a fitness filtering process to said second setto rank its members according to fitness for representing at least acorresponding portion of the series of input samples; (e) selecting atleast some of the members of the second set to form a third set; andrepeating steps (a) to (e) so as to produce said approximation sequence,wherein the third set of step (e) functions as the first set of thesubsequent step (a).
 2. A method as claimed in claim 1, wherein themethod further comprises, between steps (b) and (c), the step of: (b*)deleting one or more members of said first set.
 3. A method as claimedin claim 2, wherein step (b*) comprises removing all members of thefirst set which do not begin with the selected first n elements of step(b).
 4. A method as claimed in claim 1, wherein step (e) is conducted ina manner such as to maintain a predetermined number of members in thefirst set for each iteration.
 5. A method as claimed in claim 1, whereineach element of the approximation sequence has m possible values, mbeing an integer greater than 1, and step (c) comprises, for each memberof the first set, forming corresponding m^(n) descendent partialsequences as members of the second set by appending each possiblevalue/combination to and removing the first n elements of the member. 6.A method as claimed in claim 1, wherein each element of theapproximation sequence has m possible values, m being as integer greaterthan 1, and step (c) comprises, for each member of the first set,forming less than m^(n) corresponding descendent partial sequences asmembers of the second set by appending a limited number of possiblevalue/combinations to and removing the first n elements of the member.7. A method as claimed claim 1, wherein step (d) comprises one or moreof a group comprising low pass filtering, finite impulse responsefiltering, and recursive filtering each candidate partial sequence ofthe first set.
 8. A method as claimed in claim 1, wherein step (d)comprises giving highest precedence to members of the first set whichbegin with an element having a same sign as the next sampling point. 9.A method as claimed in claim 1, wherein the method further comprises thestep of sampling a waveform signal to obtain the series of samplevalues.
 10. A method as claimed in claim 1, wherein step (d) comprisesapplying the fitness filtering process to the members of the second setsuch as to rank them according to fitness, when appended to the previousoutput elements, for approximating the series of sample values from thefirst sample value up to and including the next sample value to beapproximated.
 11. A method as claimed in claim 1, wherein outputelements are output in the form of single-bit or multi-bit samples. 12.A system for producing a sequence approximating a series of samplevalues, the system comprising: a processing unit arranged, in use, to:(a) determine a first set having candidate partial sequences as members,each member comprising a plurality of elements; (b) select the first nelements of one of the members of the first set as a next output elementfor said approximation sequence, n being a positive integer; (c) form asecond set having descendent candidate partial sequences as members fromsaid first set; (d) apply a fitness filtering process to said second setto rank its members according to fitness for representing at least acorresponding portion of the series of input samples; (e) select atleast some of the members of the second set to form a third set; acontrol unit arranged, in use, such that the processing unit repeatssteps (a) to (e), wherein the third set of step (e) functions as thefirst set of the subsequent step (a); and an output unit arranged, inuse, to output the selected at least one output element during eachiteration, whereby the approximation sequence is produced.
 13. A systemas claimed in claim 12, wherein the processing unit is further arranged,in use, to carry out, between steps (b) and (c), the step of: (b*)deleting one or more members of said first set.
 14. A system as claimedin claim 13, wherein step (b*) comprises removing all members of thefirst set which do not begin with the selected first n elements of step(b).
 15. A system as claimed in claim 12, wherein the processing unit isarranged, in use, to conduct step (e) in a manner such as to maintain apredetermined number of members in the first set for each iteration. 16.A system as claimed in claim 12, wherein each element of theapproximation sequence has m possible values, me being an integergreater than 1, and step (c) comprises, for each member of the firstset, forming corresponding m^(n) descendent partial sequences as membersof the second set by appending each possible value/combination to andremoving the first n elements of the member.
 17. A system as claimed inclaim 12, wherein each element of the approximation sequence has mpossible values, m being an integer greater than 1, and step (c)comprises, for each member of the first set, forming less than m^(n)corresponding descendent partial sequences as members of the second setby appending a limited number of possible values/combinations to andremoving the first n elements of the member.
 18. A system as claimed inclaim 12, wherein step (d) comprises one or more of a group comprisinglow pass filtering, finite impulse response filtering, and recursivefiltering each candidate partial sequence of the first set.
 19. A systemas claimed in claim 12, wherein step (d) comprises giving highestprecedence to members of the first set which begin with an elementhaving a same sign as the next sampling point.
 20. A system as claimedin claim 12, wherein the system further comprises a sampling unit forsampling a waveform signal to obtain the series of sample values.
 21. Asystem as claimed in claim 12, wherein step (d) comprises applying thefitness filtering process to the members of the second set such as torank them according to fitness for approximating the series of samplevalues from the first sample value up to and including the next samplevalue to be approximated.
 22. A system as claimed in claim 12, whereinthe output unit is arranged, in use, such that the output elements areoutput in the form of single-bit or multi-bit samples.